Linear analysis of coupled lattices

The parametrization of linear one-dimensional optical systems using the so-called Twiss parameters, b, a, and f, has been well established for some time [1]. The extension to two dimensions involves identifying the normal modes by transforming the full turn 4 3 4 transfer matrix to block diagonal form [2]. The system is then parametrized using the six Twiss parameters for both normal modes and four additional parameters specifying the orientation of the normal modes with respect to the laboratory x, y axes. For “weak” coupling where the normal modes correspond quite closely to uncoupled horizontal and vertical motion, this analysis is conceptually simple since, to first order in the coupling, the Twiss parameters are unaffected by the coupling. However, when the coupling is strong, the situation is more complicated since the Twiss parameters now no longer have their standard meanings. For example, p b is now no longer proportional to the beam size. Furthermore, as discussed below, the identification of which mode is which at different places in the lattice can be obscured by the formalism. Since strong coupling has been put forward as a means to create round beams (for example, Talman [3] has proposed a “Möbius” twist accelerator), these subtleties need to be addressed, and this is the purpose of this paper.